It shouldn't happen to a dog, but it has -- what some in Washington might call just another White House power play. In this latest version, a new First Dog took over the presidential doghouse yesterday, scarcely a week after the first First Dog had been put out to pasture at President Reagan's California ranch.

At the White House, however, they're calling the 1-year-old, 16-pound Cavalier King Charles spaniel an early Christmas present. Reagan gave the pooch to First Lady Nancy Reagan before they left New York after an 18-hour overnight stay that included a birthday party for publisher William F. Buckley's National Review.

"Merry Christmas," Reagan told Mrs. Reagan as the pooch was brought into their Plaza Hotel suite by Kathy Osborne, the president's special assistant.

"Oh, honey, thank you, thank you, thank you," Mrs. Reagan replied, according to White House spokesman Larry Speakes.

By late yesterday, Mrs. Reagan had decided to name the dog Rex, according to Elaine Crispen, her press secretary, who added that White House chief usher Rex Scouten has announced he will leave his job soon and that this way "there's still a Rex around."

Rex, the dog, is the litter brother of Fred Buckley, one of two King Charles spaniels belonging to Buckley. White House aides said Mrs. Reagan "fell in love" with Fred after visiting the Buckleys on several occasions. Reagan secretly called the Buckleys, learned that Fred had a brother still living with the family that breeds the spaniels and made arrangements to get Rex.

Described as "housebroken and well-mannered," Rex flew with the Reagans aboard Air Force One to Andrews Air Force Base and by Marine One helicopter to the White House -- all without incident.

It was a performance that the first First Dog -- a rambunctious 65-pound bouvier named Lucky, who was known to strain at the leash and had what was delicately described as a "puddling problem" -- never quite equaled. Given the Reagans a year ago by the March of Dimes poster child, Lucky made her last public appearance a week ago at Point Mugu Naval Air Station in California, where she deplaned and promptly answered a call of nature on the tarmac. The White House announced this fall that Lucky would "retire" to California.

Speakes said yesterday Rex has been promised full access to the Oval Office, a presidential assurance with which White House Chief of Staff Donald Regan concurred.

Out of sight though Lucky is, she is not out of mind. On Thursday, Speakes said that Regan has been blamed for every departure except Lucky's, a reference to the resignation this week of national security affairs adviser Robert McFarlane.